The main goal of the study was to investigate high school advanced placement calculus teachers' subject matter and pedagogical perceptions by examining the following questions: What are the teachers' perceptions of the concept of limit, the role of limits, and the teaching of limits in calculus? Additionally, the sampling technique used shed some light on the question: Are these teachers' perceptions associated with their participation in a calculus reform project focused on staff development? A multi-case study approach involving detailed examination of six teachers (three had participated in a calculus reform project and three had not participated in any calculus reform project) was used. The data collected and analyzed included questionnaires, interviews, observational fieldnotes, videotapes of classroom instruction, journals, and written instructional documents. Upon completion of the data collection and analysis, detailed teacher profiles were created with respect to the questions above. The results of this study were then generated by searching for similarities and differences across the entire sample as well as comparing and contrasting the group of project teachers and the independent teachers. The teachers in this study perceived calculus as a linearly ordered set of topics in which the concept of limit formed the backbone for appreciating and understanding all other calculus topics. The teachers felt the intuitive understanding of limits was essential to further understanding of calculus. Nevertheless, little classtime was devoted to developing an intuitive understanding. Furthermore, little emphasis was given to drawing connections between limits and subsequent calculus topics. The independent teachers devoted considerable time to discussing formal epsilon-delta definition and arguments. The complex relationship between teachers' perceptions and classroom practice appeared to be affected by the significant influence of the teachers' goals of preparing students for the advanced placement exam and college calculus and the authority given to the calculus textbook. Differences between the group of independent teachers and the group of project teachers were found related to the following factors: (a) commitment to the textbook, (b) planning, (c) use of multiple representations, (d) attitude toward graphing technology, (e) classroom atmosphere, (f) examinations, (g) appropriate level of mathematical rigor needed for teaching calculus, and (h) the stability of perceptions. These factors, however, were not fully attributed to participation in the calculus reform project. / Graduation date: 1995

An analysis of the pertinent literature showed that there were two commonly used instructional strategies for teaching calculus to engineering technology students. Theory-problem instructional strategy is being used when an instructor first develops-the new calculus theory and then "illustrates" this theory with an applied problem. Problem-theory instructional strategy is being used when an instructor first "initiates" the new calculus theory with an applied problem and then develops the new calculus theory. The purpose of this study is to investigate which of the above instructional strategies is best for teaching calculus to engineering technology students. The evaluation was done by comparing the achievement of the theory-problem group and the problem-theory group on two tests. For the length of the study the two groups were taught identical content by the same instructor. The only difference was the order in which the applied problems were presented. There was no significant difference in achievement of the two groups on the test designed to measure understanding of techniques, principles and concepts of calculus. There was no significant difference in the achievement of the two groups on the test designed to measure success at solving applied problems. The results of this study indicate that students will do as well if they are taught by an instructional strategy which uses problems to illustrate calculus theory as they will if they are taught by an instructional strategy which uses problems to initiate the study of calculus theory. / Education, Faculty of / Graduate

The study examined classroom instructional practices and teacher's professed conceptions about teaching and learning college calculus in relationship to the implementation of scientific-programmable-graphics (SPG) calculators. The study occurred at a university not affiliated with any reform project. The participants were not the catalysts seeking to implement calculus reform, but expressed a willingness to teach the first quarter calculus course with the SPG calculator. The research design was based on qualitative methods using comparative case studies of five teachers. Primary data were collected through pre-school interviews and weekly classroom observations with subsequent interviews. Teachers' profiles were established describing general conceptions of teaching calculus, instructional practices, congruence between conceptions and practice, conceptions about teaching using SPG calculators, instructional practice with SPG calculators, and the relationship of conceptions and practice with SPG calculators. Initially, all the teachers without prior experience using SPG calculators indicated concern and skepticism about the usefulness of the technology in teaching calculus and were uncertain how to utilize the calculator in teaching the calculus concepts. During the study the teachers became less skeptical about the calculator's usefulness and found it effective for illustrating graphs. Some of the teachers' exams included more conceptual and graphically-oriented questions, but were not significantly different from traditional exams. Findings indicated the college teachers' conceptions of teaching calculus were generally consistent with their instructional practice when not constrained by time. The teachers did not perceive a dramatic change in their instructional practices. Rather, the new graphing approach curriculum and technology were assimilated into the teachers' normal teaching practices. No major shifts in the role of the teachers were detected. Two teachers demonstrated slight differences in their roles when the SPG calculators were used in class. One was a consultant to the students as they used the SPG calculators; the other became a fellow learner as the students presented different features on the calculator. Use of the calculator was influenced by several factors: inexperience with the calculator, time constraints, setting up the classroom display calculator, preferred teaching styles and emphasis, and a willingness to risk experimenting with established teaching practices and habits. / Graduation date: 1996

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This thesis describes an Action Research project within the researcher's practice as a teacher educator in Rwanda. A teaching style informed by the Theory of Didactical Situations in Mathematics (Artigue, 1994; Brousseau, 1997; 2004; Douady, 1991) and by the Zone of Proximal Development (Gallimore & Tharp, 1990; Meira & Lerman, 2001; Rowlands, 2003; Vygotsky, 1978) was conducted with first-year mathematics student teachers in Rwanda. The aim of the teaching model was to develop the student teachers' understanding of the concepts of the definite and the indefinite integrals and their link through the fundamental theorem of calculus. The findings of the analysis answer the research questions, on the one hand, of what concept images (Tall & Vinner, 1981; Vinner & Dreyfus, 1989) of the underlying concepts of integrals student teachers exhibit, and how the student teachers‟ concept images evolved during the teaching. On the other hand, the findings answer the research questions of what didactical situations are likely to further student teachers' understanding of the definite and the indefinite integrals and their link through the fundamental theorem of calculus; and finally they answer the question of what learning activities student teachers engage in when dealing with integrals and under what circumstances understanding is furthered. An analysis of student teachers' responses expressed during semi-structured interviews organised at three different points of time - before, during, and after the teaching - shows that the student teachers' evoked concept images evolved significantly from pseudo-objects of the definite and the indefinite integrals to include almost all the underlying concept layers of the definite integral, namely, the partition, the product, the sum, and the limit of a sum, especially in the symbolical representation. However, only a limited evolution of the student teachers' understanding of the fundamental theorem of calculus was demonstrated after completion of the teaching. With regard to the teaching methods, after analysis of the video recordings of the lessons, I identified nine main didactical episodes which occurred during the teaching. Interactions during these episodes contributed to the development of the student teachers' understanding of the concepts of the definite and the indefinite integrals and their link through the fundamental theorem of calculus. During these interactions, the student teachers were engaged in various cognitive processes which were purposefully framed by functions of communication, mainly the referential function, the expressive function, and the cognative function. In these forms of communication, the cognative function in which I asked questions and instructed the students to participate in interaction was predominant. The student teachers also reacted by using mainly the expressive and the referential functions to indicate what knowledge they were producing. In these exchanges between the teacher and the student teachers and among the student teachers themselves, two didactical episodes in which two student teachers overtly expressed their understanding have been observed. The analysis of these didactical episodes shows that the first student teacher's understanding has been triggered by a question that I addressed to the student after a long trial and error of searching for a mistake, whereas the second student's understanding was activated by an indicative answer given by another student to the question of the student who expressed the understanding. In the former case, the student exhibited what he had understood while in the latter case the student did not. This suggests that during interactions between a teacher and a student, asking questions further the student's understanding more than providing him or her with the information to be learnt. Finally, during this study, I gained the awareness that the teacher in a mathematics classroom has to have various decisional, organisational and managerial skills and adapt them to the circumstances that emerge during classroom activities and according to the evolution of the knowledge being learned. Also, the study showed me that in most of the time the student teachers were at the center of the activities which I organised in the classroom. Therefore, the teaching methods that I used during my teaching can assist in the process of changing from a teacher-centred style of teaching towards a student-centred style. This study contributed to the field of mathematics education by providing a mathematical framework which can be used by other researchers to analyse students' understanding of integrals. This study also contributed in providing a model of teaching integrals and of researching a mathematics (integrals) classroom which indicates episodes in which understanding may occur. This study finally contributed to my professional development as a teacher educator and a researcher. I practiced the theory of didactical situation in mathematics. I experienced the implementation of some of its concepts such as the devolution, the a didactical situation, the institutionalization, and the didactical contract and how this can be broken by students (the case of Edmond). In this case of Edmond, I realised that my listening to students needs to be improved. As a researcher, I learnt a lot about theoretical frameworks, paradigms of study and analysis and interpretation of data. The theory of didactical situations in mathematics, the action research cyclical spiral, and the revised Bloom‟s Taxonomy will remain at my hand reach during my mathematics teacher educator career. However, there is still a need to improve in the analysis of data especially from the students' standpoint; that is, the analysis of the learning aspect needs to be more practiced and improved. / Thesis (Ph.D.)-University of KwaZulu-Natal, Pietermaritzburg, 2010.

In the first part of the project the elementary development of an extended number system called Hyperreals is discussed. The second half of this project develops the basics of Nonstandard Analysis, including the theory of ultrafilters, and the formal construction of the Hyperreals.

<p>The purpose of this study was to investigate the epistemological obstacles that mathematics students at undergraduate level encounter in coming to understand the limit concept. The role played by language and symbolism in understanding the limit concept was also investigated. A group of mathematics students at undergraduate level at the National University of Lesotho (NUL) was used as the sample for the study. Empirical data were collected by using interviews and questionnaires. These data were analysed using both the APOS framework and a semiotic perspective.</p> <p><br /> Within the APOS framework, the pieces of knowledge that have to be constructed in coming to understand the limit concept are actions, processes and objects. Actions are interiorised into processes and processes are encapsulated into objects. The conceptual structure is called a schema. In investigating the idea of limit within the context of a function some main epistemological obstacles that were encountered when actions were interiorised into processes are over-generalising and taking the limit value as the function value. For example, in finding the limit value L for f(x) as x tends to 0, 46 subjects out of 251 subjects said that they would calculate f(0) as the limit value. This method is appropriate for calculating the limit values for continuous functions. However, in this case, the method is generalised to all the functions. When these subjects encounter situations in which the functional value is equal to the limit value, they take the two to be the same. However, the two are different entities conceptually.</p>

Due in part to the growing popularity of the Advanced Placement program, an increasingly large percentage of entering college students are enrolling in calculus courses having already taken calculus in high school. Many students do not score high enough on the AP calculus examination to place out of Calculus I, and many do not take the examination. These students take Calculus I in college having already seen most or all of the material. Students at two colleges were surveyed to determine whether prior calculus experience has an effect on these students' effort levels or motivation. Students who took calculus in high school did not spend as much time on their calculus coursework as those who did not take calculus, but they were just as motivated to do well in the class and they did not miss class any more frequently. Prior calculus experience was not found to have a negative effect on student motivation or effort. Colleges should work to ensure that all students with prior calculus experience receive the best possible placement, and consider making a separate course for these students, if it is practical to do so.

Ball State University LibrariesLibrary services and resources for knowledge buildingMasters ThesesThere is no abstract available for this thesis.

<p>The purpose of this study was to investigate the epistemological obstacles that mathematics students at undergraduate level encounter in coming to understand the limit concept. The role played by language and symbolism in understanding the limit concept was also investigated. A group of mathematics students at undergraduate level at the National University of Lesotho (NUL) was used as the sample for the study. Empirical data were collected by using interviews and questionnaires. These data were analysed using both the APOS framework and a semiotic perspective.</p> <p><br /> Within the APOS framework, the pieces of knowledge that have to be constructed in coming to understand the limit concept are actions, processes and objects. Actions are interiorised into processes and processes are encapsulated into objects. The conceptual structure is called a schema. In investigating the idea of limit within the context of a function some main epistemological obstacles that were encountered when actions were interiorised into processes are over-generalising and taking the limit value as the function value. For example, in finding the limit value L for f(x) as x tends to 0, 46 subjects out of 251 subjects said that they would calculate f(0) as the limit value. This method is appropriate for calculating the limit values for continuous functions. However, in this case, the method is generalised to all the functions. When these subjects encounter situations in which the functional value is equal to the limit value, they take the two to be the same. However, the two are different entities conceptually.</p>